Question

How do you write the function in standard form $y = {\left( {x + 3} \right)^2} + 2$?

Answer 1

Hint: A quadratic equation is a second-degree polynomial equation having a standard form of $a{x^2} + bx + c$. Here a and b are coefficients of the variable and c is the constant. Hence a quadratic equation is $f\left( x \right) = y = a{x^2} + bx + c$. When we substitute an input x in the function, it will give an output y.

Complete step by step solution:
The given equation is $y = {\left( {x + 3} \right)^2} + 2$. And we know the standard form of a quadratic equation which is written as
$ \Rightarrow y = a{x^2} + bx + c$
Let us now rearrange the given equation to get the required quadratic form.
$ \Rightarrow y = {\left( {x + 3} \right)^2} + 2$
We know that ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
Therefore ${\left( {x + 3} \right)^2}$ in the above equation can be written as
$ \Rightarrow y = {x^2} + 6x + 9 + 2$
Add $9$ and $2$, we get
$ \Rightarrow y = {x^2} + 6x + 11$
Where the coefficients $a = 1,b = 6$ and the constant $c = 11$.
The above equation is the standard form of a quadratic function and it is also known as a second-degree polynomial function.

Therefore, $y = {x^2} + 6x + 11$ is the required standard form of given function.

Note: The quadratic equation is a very important form of an equation that is frequently seen in algebraic problems. Sometimes a problem may require finding a quadratic equation using its roots alone. In that case, we should form an equation containing the roots complemented by a factor. For example, if one of the roots is 3, the corresponding factor should be $\left( {x - 3} \right)$ which is equal to zero. By multiplying these factors, we can find out the quadratic equation.